Clearly, we have \(|S_n|=|S|=\prod _{i=1}^n|X_i|\) . With a slight abuse of notation, we identify \(S_n\)  with \(S\) .
The set system \(\cup _{k=0}^{n}S_k\)  is an abstract simplicial complex, that is, for each of its elements \(A\) , every subset of \(A\)  also belongs to \(\cup _{k=0}^{n}S_k\) . This simplicial complex has a geometric realization in \(\mathbb {R}^{2n-1}\) , where every element \(A\)  is represented by an \((|A|-1)\) -dimensional simplex. (See [1]}, part II, Section 9 or [2]}, Section 1.5. Note that not all textbooks consider the empty set a \(-1\) -dimensional simplex, but we do.)
